Why are inhibitory connections often used in virtual neural networks when they don't seem to exist in real life neural networks?
I think you are confused. The reason why some neural neural networks have neurons with an output in range (-1, 1) all depends on the activation function used. Some networks even have neurons with an output range of (-Infinity, +Infinity) ( aka the identity function ).

I advise you to take a look at this list: activation functions w/ ranges. 

Nowadays a lot of neural networks have a mixture of different activation functions. E.g. combining ReLU with Sigmoid or TanH. 

This neural network library even shows that for most simple problems, neural networks prefer a variety of activation functions. 

The red lines just mean that that connection has a low value in comparison with other connections.

There is no rule of thumb for activation functions, some problems benefit from a certain function, some don't.